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Resolución de integrales/ (x^3)/ i*n*x/ sin(pi*n*x/l)*t*(x^3)*l^2

Integral de sin(pi*n*x/l)*t*(x^3)*l^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                       
  /                       
 |                        
 |     /pi*n*x\    3  2   
 |  sin|------|*t*x *l  dx
 |     \  l   /           
 |                        
/                         
0
∫01l2x3tsin⁡(xπnl) dx\int\limits_{0}^{1} l^{2} x^{3} t \sin{\left(\frac{x \pi n}{l} \right)}\, dx 
Integral(((sin(((pi*n)*x)/l)*t)*x^3)*l^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                     /    //                                      0                                        for n = 0\                                   \
                                     |    ||                                                                                        |                                   |
                                     |    ||   //     3    /pi*n*x\      2    /pi*n*x\        2    /pi*n*x\            \            |                                   |
                                     |    ||   ||  2*l *sin|------|   l*x *sin|------|   2*x*l *cos|------|            |            |                                   |
  /                                  |    ||   ||          \  l   /           \  l   /             \  l   /            |            |      //       0         for n = 0\|
 |                                   |    ||   ||- ---------------- + ---------------- + ------------------  for n != 0|            |      ||                          ||
 |    /pi*n*x\    3  2             2 |    ||   ||         3  3              pi*n                 2  2                  |            |    3 ||      /pi*n*x\            ||
 | sin|------|*t*x *l  dx = C + t*l *|- 3*|<-l*|<       pi *n                                  pi *n                   |            | + x *|<-l*cos|------|            ||
 |    \  l   /                       |    ||   ||                                                                      |            |      ||      \  l   /            ||
 |                                   |    ||   ||                             3                                        |            |      ||---------------  otherwise||
/                                    |    ||   ||                            x                                         |            |      \\      pi*n                /|
                                     |    ||   ||                            --                              otherwise |            |                                   |
                                     |    ||   \\                            3                                         /            |                                   |
                                     |    ||-----------------------------------------------------------------------------  otherwise|                                   |
                                     \    \\                                     pi*n                                               /                                   /
∫l2x3tsin⁡(xπnl) dx=C+l2t(x3({0for n=0−lcos⁡(πnxl)πnotherwise)−3({0for n=0−l({−2l3sin⁡(πnxl)π3n3+2l2xcos⁡(πnxl)π2n2+lx2sin⁡(πnxl)πnfor n≠0x33otherwise)πnotherwise))\int l^{2} x^{3} t \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + l^{2} t \left(x^{3} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \cos{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \left(\begin{cases} - \frac{2 l^{3} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi^{3} n^{3}} + \frac{2 l^{2} x \cos{\left(\frac{\pi n x}{l} \right)}}{\pi^{2} n^{2}} + \frac{l x^{2} \sin{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{for}\: n \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)\right)
Simplificar
Respuesta [src] 
/     /       /pi*n\      4    /pi*n\      2    /pi*n\      3    /pi*n\\                                  
|     |  l*cos|----|   6*l *sin|----|   3*l *sin|----|   6*l *cos|----||                                  
|   2 |       \ l  /           \ l  /           \ l  /           \ l  /|                                  
|t*l *|- ----------- - -------------- + -------------- + --------------|  for And(n > -oo, n < oo, n != 0)
<     |      pi*n            4  4             2  2             3  3    |                                  
|     \                    pi *n            pi *n            pi *n     /                                  
|                                                                                                         
|                                   0                                                otherwise            
\
{l2t(−6l4sin⁡(πnl)π4n4+6l3cos⁡(πnl)π3n3+3l2sin⁡(πnl)π2n2−lcos⁡(πnl)πn)for n>−∞∧n<∞∧n≠00otherwise\begin{cases} l^{2} t \left(- \frac{6 l^{4} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{4} n^{4}} + \frac{6 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{3 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right) & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
=
= 
/     /       /pi*n\      4    /pi*n\      2    /pi*n\      3    /pi*n\\                                  
|     |  l*cos|----|   6*l *sin|----|   3*l *sin|----|   6*l *cos|----||                                  
|   2 |       \ l  /           \ l  /           \ l  /           \ l  /|                                  
|t*l *|- ----------- - -------------- + -------------- + --------------|  for And(n > -oo, n < oo, n != 0)
<     |      pi*n            4  4             2  2             3  3    |                                  
|     \                    pi *n            pi *n            pi *n     /                                  
|                                                                                                         
|                                   0                                                otherwise            
\
{l2t(−6l4sin⁡(πnl)π4n4+6l3cos⁡(πnl)π3n3+3l2sin⁡(πnl)π2n2−lcos⁡(πnl)πn)for n>−∞∧n<∞∧n≠00otherwise\begin{cases} l^{2} t \left(- \frac{6 l^{4} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{4} n^{4}} + \frac{6 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{3 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right) & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases} 
Piecewise((t*l^2*(-l*cos(pi*n/l)/(pi*n) - 6*l^4*sin(pi*n/l)/(pi^4*n^4) + 3*l^2*sin(pi*n/l)/(pi^2*n^2) + 6*l^3*cos(pi*n/l)/(pi^3*n^3)), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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